We now consider a hydrogen atom in equilibrium with a heat reservoir at a temperature \(\tau \) . Pertinent to the hyperfine structure of an H atom at such thermal equilibrium is the density operator \(\rho (\tau )=\frac{1}{Z}e^{-\beta {H_{HF}}}\) , where \(Z=Tr(e^{-\beta {H_{HF}}})\)  is the partition function, \(\beta =1/(k_B\tau )\) , and \(k_B\)  is the Boltzmann constant. While quantum entanglement in pure quantum states is adequately understood in terms of the von Neumann entropy [1]}, [2]}, a suitable quantifier for the entanglement of mixed states is much harder to define. Yet, the entanglement of formation has been shown to satisfactorily address this problem [3]}, [4]}, providing a closed-form computable measure for the entanglement of mixed states, fully applicable to thermal states as defined by the density operator \(\rho (\tau )\)  [5]}, [6]}. Specifically, given \(\rho \)  = \(\rho (\tau )\) , the entanglement of such states is meaningfully quantified via the concurrence [3]}, [4]} \( C=\textrm {max}\lbrace 0,\lambda _1-\lambda _2-\lambda _3-\lambda _4\rbrace \) , where \(\lambda _i\) , enumerated such that, \(\lambda _1\ge \lambda _2\ge \lambda _3\ge \lambda _4\) , are the eigenvalues of the Hermitian matrix \(R=\sqrt{\sqrt{\rho }\tilde{\rho }\sqrt{\rho }}\)  and \(\tilde{\rho }=(\sigma _y\otimes \sigma _y)\rho ^*(\sigma _y\otimes \sigma _y)\) .
